A Numerical Investigation of Multiple Traveling Pulse and Front Solutions

ABSTRACT Algorithms are proposed to calculate traveling pulses and fronts in both directions for the FitzHugh- Nagumo equations in one dimensional spatial domain. The rst algorithm is based on the application of the steepest descent method to a certain functional on some admissible sets. These sets are dierent for pulses and for fronts. This approach is global in nature, so that an initial guess for the wave prole and the speed can be quite dierent from the correct solution. The second algorithm is the pseudo arc length continuation method, which solves the governing equations directly. The two algorithms are complementary. Continuation makes the computation of a bifurcation diagram more ecient, but it requires a good initial guess. This is supplied by the steepest descent algorithm. Also, the two algorithms serve as an independent check for one another. Depending on the physical parameter values, we observe the existence of single, multiple (stable and unstable) or no traveling pulses and fronts, within the corresponding admissible set. At suitable parameter values, we found as many as ve traveling wave solutions; two distinct pulses and two fronts moving to the right, and one front moving to the left. The computed wave proles are tested numerically using a parabolic solver and, for stable solutions, the speed and shape are maintained very well for a large number of time steps